138 XIII INTEGRATION Now the sets Apt9tf are closed (3.11.4), hence BM?r is integrable (13.8.7). More- over, by hypothesis, for each integer r ^ 1 we have ^/f) Bw>r\ = 0, and the sequence (B^Xg! is decreasing, so that by (13.8.7) we have lim yu(Bn>r) = 0. n-»oo Hence there exists an integer nr such that n(Bnrtf) g s/2r+2. Let B = (J BMr,r; r the set B is integrable, and we have //(B) g f] s/2r+2 = Je by (13.8.7). The r=l set C = K0 - B is also integrable, and by definition the sequence (fn \ C) converges uniformly to f\ C. Hence we may take K' to be any compact subset of C such that n(C - K') ^ & (13.7.9). (13.9.11) Let (fn) be any sequence of measurable functions on X with values in R. Then the functions inffn> sup/M, lim sup/M and lim inffn are measurable. n n ./i-* oo /i-* oo For sup/, is the limit of the increasing sequence of functions gn = sup ft, n l£i£n which are measurable (13.9.7); and lim sup/n is the limit of the decreasing w* oo sequence of functions sup/n+p, which are measurable, by what has just been p*o proved. (184.108.40.206) The propositions (13.9.6), (13.9.7), (13.9.9), and (13.9.11) remain valid when the word "measurable" is replaced throughout by "universally measurable." On the other hand, a function cannot be universally measurable unless it is everywhere defined, because any nonempty subset N of X has measure //(N) ^ 0 if for example \JL is the Dirac measure at a point of N. In particular (12.7.8), every function belonging to«/ (resp. ff1) is universally measurable. If / is any lower semicontinuous function on X, then by writing / = inf (sup (/, - n)} and sup (/, - n) = ~ n + sup (/ + n, 0) n and using (13.9.6), it follows that (13L9.11.2) Every (upper or lower) semicontinuous function on X with values in R is universally measurable. The same will be true of all functions obtained from semi-continuous functions by repeated applications of the operations in (13.9.6), (13.9.7), and (13.9.11).ictions u \ Ln is continuous.